Computational Complexity on the Blackboard

The theory of computational complexity is based on the tradeoff between two computational resources, memory space and computer time. This paper investigates the physical counterparts of these resources. Memory space is the number of bits of information required by a computer over the course of a computation. Physically, information is intimately connected with entropy: entropy can be thought of as random bits of information. The physical analogue of the amount of memory space required for a computation, Ф- SPACE, is the difference between the maximum entropy Smax of the computing system and its actual entropy, S.Ф-SPACE = S - Smax is the thermodynamic depth of the computation. Ф-SPACE is the maximum number of bits of memory space available for a physical system to perform computation. In conventional computational complexity theory, computer time is identified as the number of elementary logical operations (AND, OR, NOT, COPY) required to perform the computation. The physical analogue of time computational complexity is simply the time required for the physical computing system to perform the computation. In order to obtain a number that measures the number of physical operations that can be performed by the system over time t, we define Ф-TIME to be proportional to the computation time multiplied by the energy E of the computing system over the course of the computation: Ф-TIME = 2Et/πħ. Here, the proportionality factor 2/πħ is chosen to match the time it takes a computing system with energy E to flip a bit in time t. Ф-TIME is thus the number of bit flips that can take place in an information processing system. In addition, we propose a novel measure of physical complexity that combines these measures. The free energy F = E - TS of a system is the amount of energy that is available to do useful work - as opposed to the energy bound up in the thermal fluctuations of heat. We define the free complexity to be the free energy × time product, 2Ft/πħ required by a physical system to perform a computation. Physical spatial and temporal complexity provide upper bounds on the amount of computation that can be performed by any physical system. We discuss the implications of physical complexity for the computational complexity hierarchy, in particular, we show that in the presence of errors in the computation, Ф-SPACE is proportional Ф-TIME. We apply the resulting theory of physical computational complexity to analyze the computational power of black holes: exactly half of the energy of a black hole is available for physical computation.

Wolfram's Cellular Automata and Complexity, an anthology of some of his best work in the area, was published in 1994, and it seems fitting to attempt to explain how his seminal work can be put to good use in laboratory‐based computer science education. This article documents the advantages of using cellular automata (CA) to model complexity theory, chaos, and some aspects of the theory of computation. It explains how CA are constructed, it details the fourfold classification scheme advocated by Wolfram, and it explores in some depth issues associated with induction and Class 2 CA. Several CA are examined with an eye to determining what kinds of empirical evidence count for and against such classifications. The argument is that such lab explorations are pedagogically effective and engaging as an introduction to computer modeling, complexity theory, and related issues in the theory of computation. Software to run the CA simulations has been written by the author and can be downloaded from America Online and...

THE INTERFACE BETWEEN LANGUAGE THEORY AND COMPLEXITY THEORY

Kauffman nets, also known as N-K models, have been studied extensively because their dynamics can be used to model a variety of interesting dynamical processes. This paper investigates the properties of the problem of determining whether or not a given configuration of a Kauffman net has a predecessor. Here it is shown that when the parameter K that governs the number of connections grows as ln(N) , where N is the number of elements, the problem of finding a solution is extremely sensitive to small changes in the problem statement. This result has implications for studies of the physics of random systems and also may have applications for questions in computational complexity theory.

The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. In particular, the complexities of several fundamental problems in constrained block coding are precisely classified in terms of the existing complexity-theoretic structure. One type of problem studied is that of designing encoder and decoder circuits using minimum or approximately minimum hardware; for our purposes, an "input" to this problem is (i) a deterministic, irreducible finite-state transition diagram (DIF) defining a set of constrained binary sequences, and (ii) a desired rate p:q. Several of these minimum-encoder and minimum-decoder problems are shown to be NP-hard, and more interestingly some are shown to be complete in the second and third levels of the polynomial hierarchy. Another fundamental problem is that of computing the maximum rate of a block code; that is, given a DIF and a codeword length q, find the maximum p such that a rate p:q block code exists for the constraint defined by the DIF. This problem is shown to be NP/sup #P/-complete. Although it is not known whether NP/sup #P/ contains problems of super-polynomial complexity, it lies "higher" in the complexity-class structure than NP in the sense that it is possible, given current knowledge, that NP/sup #P/ contains problems of super-polynomial complexity even if P=NP. Another question studied is whether maximum rate block codes can always be implemented by encoders and decoders of polynomial size. The answer to this question is shown to be closely related to whether the class #P lies "lower" in the complexity-class structure than currently believed-a proof of either answer would have major implications in complexity theory.

Herbert A. Simon and Alan Newell won the Turing Award jointly in Computer Science for foundational work on Artificial Intelligence. Simon also won the Nobel Prize in Economics for the concept of “bounded rationality.” In both cases, the same heuristic was deemed fundamental: “Search till a satisfactory solution is found.” We argue that behavioral strategy can learn a great deal from the Theory of Computational Complexity and Artificial Intelligence. These fields can provide a sounder theoretical grounding for bounded rationality and for the necessity and usefulness of heuristics. Finally, a concept of “organizational intractability” based roughly on the metaphor provided by the Theory of Computational Complexity may be useful in determining what analytical decision technologies are actually intractable in real organizations with constraints on time and managerial attention.

This special issue of Theory of Computing Systems contains articles which relate to talks given at the 8th International Conference on Computability, Complexity and Randomness (CCR) in Moscow from the 23rd to the 27th of September 2015. The CCR conference series is devoted to algorithmic randomness, Kolmogorov complexity, and their relationship with computability theory, complexity theory, logic and reverse mathematics. The contributions to this special issue reflect both the interests of reserachers in Russia, in particular Kolmogorov and computational complexity, and new directions in the field of algorithmic randomness such as the focus on normal numbers and connections to ergodic theory. No conference proceedings have been published; this special issue is the only publication relating to the confernece. All contributions are a significant extension of the conference presentations, and underwent the usual rigorous journal reviewing process. The article Polynomial-time algorithms for checking some properties of Boolean functions given by polynomials by Svetlana Selezneva and Anton Bukhman presents a polynomial time algorithm, which tests whether a Boolean function represented by a multivariate polynomial over GF(2) is even, that is, has zero derivative in the direction of (1, . . . , 1). The approach is based on the notion of the characteristic of monomials in the polynomial representation of Boolean functions. In Normality and finite-state dimension of Liouville numbers Satyadev Nandakumar gives a new construction of a Liouville number which is normal for a given base. Vladimir V’yugin in On Stability of Probability Laws with Respect to Small Violations of Algorithmic Randomness examines the robustness of ergodic theorems when the assumption of randomness is relaxed only a little. For example he shows that

Morphological computation proposes the idea that in a physical system, certain computational processes can be off-loaded to the body. However, the concept has still eluded serious theoretical quantification attempts, unlike traditional computational theory. This perspective examines the notion of morphological computation from the well established theories of traditional computation and computational complexity, drawing parallels between the two, to understand the differences and similarities. Further, we look at the quantification efforts of morphological computation and attempt to link it to the unexplored field of control complexity. We argue that the development of complexity theory for control problems is necessary to study and utilize the concept of morphological computation, if it is possible.

The notion of NP-completeness has helped researchers in several fields to argue that some problems are intrinsically difficult to solve computationally, even though proof of that is beyond the current state of our knowledge. The study of properties of complete sets and reducibilities has led to the development of a branch of complexity theory called structural complexity theory. This paper is a brief survey of recent results in structural complexity theory and is aimed at researchers who know something about computational complexity theory but do not work in structural complexity theory. To describe the theory in a way that gives the reader some concrete applications, the results are described in terms of their relationship to P =?NP and other problems.

Approximate counting with uniform constant-depth circuits by M. Ajtai On strong separations from $AC^0$ by E. Allender and V. Gore Parallel matching complexity of Ramsey's theorem by J. Beck On algorithms for simple stochastic games by A. Condon Locally random reductions in interactive complexity theory by J. Feigenbaum An application of game-theoretic techniques to cryptography by M. J. Fischer and R. N. Wright Composition of the universal relation by J. Ha stad and A. Wigderson Practical perfect cryptographic security by U. M. Maurer Fair games against an all-powerful adversary by R. Ostrovsky, R. Venkatesan, and M. Yung Factoring integers and computing discrete logarithms via diophantine approximation by C. P. Schnorr A new lower bound theorem for read-only-once branching programs and its applications by J. Simon and M. Szegedy On the E -isomorphism problem by J. Wang.

The object of study of this paper is the following multi-determinantal algebraic variety, $\text{SING}_{n, m}$ , which captures the symbolic determinant identity testing (SDIT) problem (a canonical version of the polynomial identity testing (PIT) problem), and plays a central role in algebra, algebraic geometry and computational complexity theory. $\text{SING}_{n, m}$ is the set of all $m$ -tuples of $n\times n$ complex matrices which span only singular matrices. In other words, the determinant of any linear combination of the matrices in such a tuple vanishes. The algorithmic complexity of testing membership in $\text{SING}_{n, m}$ is a central question in computational complexity. Having almost a trivial probabilistic algorithm, finding an efficient deterministic algorithm is a holy grail of derandomization, and to top it, will imply super-polynomial circuit lower bounds! A sequence of recent works suggests efficient deterministic “geodesic descent” algorithms for memberships in a general class of algebraic varieties, namely the null cones of (reductive) linear group actions. Can such algorithms be used for the problem above? Our main result is negative: $\text{SING}_{n, m}$ is not the null cone of any such group action! This stands in stark contrast to a non-commutative analog of this variety (for which such algorithms work), and points to an inherent structural difficulty of $\text{SING}_{n, m}$ . In other words, we provide a barrier for the attempts of derandomizing SDIT via these algorithms. To prove this result we identify precisely the group of symmetries of $\text{SING}_{n, m}$ . We find this characterization, and the tools we introduce to prove it, of independent interest. Our characterization significantly generalizes a result of Frobenius for the special case $m=1$ (namely, computing the symmetries of the determinant). Our proof suggests a general method for determining the symmetries of general algebraic varieties, an algorithmic problem that was hardly studied and we believe is central to algebraic complexity.

In the framework of Membrane Computing, several tools to tackle the P versus NP problems by means of frontiers of the efficiency expressed in terms of syntactic or semantic ingredients, have been developed. In this paper, an overview of the results in computational complexity theory concerning to membrane systems (tissue-like and cell-like approach) with symport/antiport rules (where objects are transported without evolving), is given. The frontiers are formulated regarding the length of communication rules, the kind of rules implementing the production of an exponential number of cells/membranes in polynomial time, and the role of the environment. An interesting remark of the obtained results refers that the underlying structure to membrane systems (directed graph versus rooted tree) does not matter in this context.

Computability and computational complexity were first considered over the fields of real and complex numbers and generalized to arbitrary algebraic systems. This article approaches the theory of computational complexity over an arbitrary algebraic system by taking computability over the list extension for a computational model of it. We study the resultant polynomial complexity classes and mention some NP-complete problems.

Similar to algorithms, which consume time and memory to run, hardware requires resources to function. For devices processing physical waves, implementing operations needs sufficient "space," as dictated by wave physics. How much space is needed to perform a certain function is a fundamental question in optics, with recent research addressing it for given mathematical operations, but not for more general computing tasks, e.g., classification. Inspired by computational complexity theory, we study the "spatial complexity" of optical computing systems in terms of scaling laws-specifically, how their physical dimensions must scale as the dimension of the mathematical operation increases-and propose a new paradigm for designing optical computing systems: space-efficient neuromorphic optics, based on structural sparsity constraints and neural pruning methods motivated by wave physics (notably, the concept of "overlapping nonlocality"). On two mainstream platforms, free-space optics and on-chip integrated photonics, our methods demonstrate substantial size reductions (to 1%-10% the size of conventional designs) with minimal compromise on performance. Our theoretical and computational results reveal a trend of diminishing returns on accuracy as structure dimensions increase, providing a new perspective for interpreting and approaching the ultimate limits of optical computing-a balanced trade-off between device size and accuracy.

Computational and complexity theory are core components of the computer science curriculum, and in the vast majority of cases are taught using decision problems as the main paradigm. For experienced practitioners, decision problems are the best tool. But for undergraduates encountering the material for the first time, we present evidence that non-decision problems (such as optimization problems and search problems) are preferable. In addition, we describe technical definitions and pedagogical strategies that have been used successfully for teaching the theory course using non-decision problems as the central concept.